We have a Pyro equipped with the Degreaser, Flare gun , and Axtinguisher.

This Pyro uses airblasts constantly and effectively, has near perfect flare gun aim consistently getting airshots, is a master of the Puff and Sting, Spy checks, extinguishes team mates, and always captures points/intel.

We have another Pyro with the Backburner, Shotgun, and Homewrecker.

This Pyro only airblasts when there are crit projectiles or if you are about to die from afterburn, knows how to flank and ambush, has perfect shotgun aim and always uses it at the right time, puts the homewrecker to good use removing sappers, and finishing off engineer nests, and always captures points/intel.

Ambushing:
Pyro 1 can ambush equally as well as Pyro 2 due to the axtinguisher. Tie.

Combat:
Pyro #1 is better against heavy classes and 1v1 due to his reflections and controlled crits. Pyro #2's dps and shotgun reliability make him better against weaker classes, and his buffed flamethrower is good against groups. Tie.

Support:
Both are spychecking, both are eliminating strong classes through separate means, but pyro 2 is destroying sappers (which the engi should be doing anyways).
Tie.

End result:
They both play differently and both are stronger in their own situations, but they are both equally useful.

*Epic slow-motion crash through the window*
*Lands on feet, bows, and sternly addresses the room*

Ahem. it seems to me that I have competition. Pardon me, but allow me to show you this:

The Festive Flamethrower
A merrymaking device capable of inflicting horrible and permanently-disfiguring injuries, but charming and jolly enough to prevent oneself from losing their mind. Comes in Red-Yellow-Green and Blue-Yellow-Green varieties.

The Shotgun
This old classic never fails to meet the mark. With a prominence in firepower and a reliable feeding system, you can't go wrong taking this with you.

The Maul
A relic from the future, this Marsian hydraulics-powered hand-powered machinery-smashing device allows for the upmost highest efficiency in demolishing anything you see fit.

*A dramatic pause as the camera zooms in on the mysterious gentleman's face*

This has been Admats, the ninja salesman.

*A puff of smoke and he's gone, leaving behind a business card covered in blood and an unknown energy drink*

Pyro 1 is more deadly, and can protect his teammates against spam in the frontiles easily, also can pick up targets faster due to not only having Degreaser + Axetinguisher but also Flare Gun, making him able to kill Medics at range

Pyro 2 can do a good job at defense, and killing lone targets, as well as protect his Engie, but if the time is needed he can't outstall ubers or reflect too many projectiles, he'll have a hard time against an organizated team, basically the Backburner limits yourself if you want to be supportive, basically the first Pyro can do everything the second can, except for removing sappers, but by Spycheking you're helping your Engie more than enough (who should be able to deal with a single Spy by it's own)

Backburner Pyros are more credit to team by killing people (aka slowing down the enemy team) than defending them, so they are almost equal, except the first can also go around killing with the same efficiency

The winner is the demoman or soldier who can spend half the effort and get twice the results of either of the pyros. Due to soldier and demo being well designed and not balanced around constantly switching weapons for no real reason other than a low-damage crit.